LMFDB - Embedded newform 116.6.c.a.57.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,6,Mod(57,116)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("116.57");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 116 = 2^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	116.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.6045230983$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 1783 x^{10} + 1098222 x^{8} + 308936830 x^{6} + 41608294477 x^{4} + 2492454459339 x^{2} + 53357557623300 $ x^12 + 1783x^10 + 1098222x^8 + 308936830x^6 + 41608294477x^4 + 2492454459339*x^2 + 53357557623300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			57.12
Root			$29.4895i$ of defining polynomial
Character	$\chi$	$=$	116.57
Dual form			116.6.c.a.57.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+29.4895i q^{3} +28.0043 q^{5} -60.4647 q^{7} -626.633 q^{9} +O(q^{10})$	$q+29.4895i q^{3} +28.0043 q^{5} -60.4647 q^{7} -626.633 q^{9} +86.9608i q^{11} -488.444 q^{13} +825.834i q^{15} +647.445i q^{17} -426.146i q^{19} -1783.08i q^{21} +53.1312 q^{23} -2340.76 q^{25} -11313.1i q^{27} +(2955.04 - 3432.04i) q^{29} -3369.70i q^{31} -2564.43 q^{33} -1693.27 q^{35} +8408.75i q^{37} -14404.0i q^{39} +9422.03i q^{41} -14712.3i q^{43} -17548.4 q^{45} +14066.5i q^{47} -13151.0 q^{49} -19092.8 q^{51} -29222.8 q^{53} +2435.28i q^{55} +12566.8 q^{57} +24544.2 q^{59} +29241.9i q^{61} +37889.1 q^{63} -13678.5 q^{65} -11066.9 q^{67} +1566.81i q^{69} +59435.2 q^{71} -38646.0i q^{73} -69027.9i q^{75} -5258.06i q^{77} +45902.2i q^{79} +181348. q^{81} -118953. q^{83} +18131.2i q^{85} +(101209. + 87142.9i) q^{87} +65646.3i q^{89} +29533.6 q^{91} +99370.9 q^{93} -11933.9i q^{95} +132472. i q^{97} -54492.4i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9}+O(q^{10}) $ 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9	$ 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9} - 194 q^{13} + 3996 q^{23} + 1782 q^{25} - 1900 q^{29} + 7266 q^{33} + 12348 q^{35} - 13988 q^{45} - 8396 q^{49} - 10788 q^{51} - 28626 q^{53} + 11140 q^{57} + 27356 q^{59} + 55392 q^{63} - 55126 q^{65} - 13184 q^{67} + 44352 q^{71} + 74264 q^{81} - 192628 q^{83} + 98908 q^{87} + 13580 q^{91} + 59710 q^{93}+O(q^{100}) $ 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9 - 194 * q^13 + 3996 * q^23 + 1782 * q^25 - 1900 * q^29 + 7266 * q^33 + 12348 * q^35 - 13988 * q^45 - 8396 * q^49 - 10788 * q^51 - 28626 * q^53 + 11140 * q^57 + 27356 * q^59 + 55392 * q^63 - 55126 * q^65 - 13184 * q^67 + 44352 * q^71 + 74264 * q^81 - 192628 * q^83 + 98908 * q^87 + 13580 * q^91 + 59710 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/116\mathbb{Z}\right)^\times$.

$n$	$59$	$89$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			29.4895i			1.89175i	0.324525	+	0.945877i	$0.394796\pi$
$3$			29.4895i			1.89175i	−0.324525	+	0.945877i	$0.605204\pi$
$4$		0			0
$5$	28.0043			0.500956			0.250478	−	0.968122i	$-0.419412\pi$
$5$	28.0043			0.500956			0.250478	+	0.968122i	$0.419412\pi$
$6$		0			0
$7$	−60.4647			−0.466398			−0.233199	−	0.972429i	$-0.574919\pi$
$7$	−60.4647			−0.466398			−0.233199	+	0.972429i	$0.574919\pi$
$8$		0			0
$9$	−626.633			−2.57873
$10$		0			0
$11$			86.9608i			0.216691i	0.994113	+	0.108346i	$0.0345553\pi$
$11$			86.9608i			0.216691i	−0.994113	+	0.108346i	$0.965445\pi$
$12$		0			0
$13$	−488.444			−0.801597			−0.400799	−	0.916166i	$-0.631267\pi$
$13$	−488.444			−0.801597			−0.400799	+	0.916166i	$0.631267\pi$
$14$		0			0
$15$			825.834i			0.947686i
$16$		0			0
$17$			647.445i			0.543351i	0.962389	+	0.271675i	$0.0875777\pi$
$17$			647.445i			0.543351i	−0.962389	+	0.271675i	$0.912422\pi$
$18$		0			0
$19$		−	426.146i		−	0.270816i	−0.990790	−	0.135408i	$-0.956765\pi$
$19$		−	426.146i		−	0.270816i	0.990790	−	0.135408i	$-0.0432345\pi$
$20$		0			0
$21$		−	1783.08i		−	0.882311i
$22$		0			0
$23$	53.1312			0.0209426			0.0104713	−	0.999945i	$-0.496667\pi$
$23$	53.1312			0.0209426			0.0104713	+	0.999945i	$0.496667\pi$
$24$		0			0
$25$	−2340.76			−0.749043
$26$		0			0
$27$		−	11313.1i		−	2.98658i
$28$		0			0
$29$	2955.04	−	3432.04i	0.652483	−	0.757804i
$29$	2955.04	−	3432.04i	0.652483	−	0.757804i
$30$		0			0
$31$		−	3369.70i		−	0.629777i	−0.949129	−	0.314889i	$-0.898033\pi$
$31$		−	3369.70i		−	0.629777i	0.949129	−	0.314889i	$-0.101967\pi$
$32$		0			0
$33$	−2564.43			−0.409927
$34$		0			0
$35$	−1693.27			−0.233645
$36$		0			0
$37$			8408.75i			1.00978i	0.863183	+	0.504891i	$0.168467\pi$
$37$			8408.75i			1.00978i	−0.863183	+	0.504891i	$0.831533\pi$
$38$		0			0
$39$		−	14404.0i		−	1.51643i
$40$		0			0
$41$			9422.03i			0.875356i	0.899132	+	0.437678i	$0.144199\pi$
$41$			9422.03i			0.875356i	−0.899132	+	0.437678i	$0.855801\pi$
$42$		0			0
$43$		−	14712.3i		−	1.21341i	−0.794927	−	0.606706i	$-0.792491\pi$
$43$		−	14712.3i		−	1.21341i	0.794927	−	0.606706i	$-0.207509\pi$
$44$		0			0
$45$	−17548.4			−1.29183
$46$		0			0
$47$			14066.5i			0.928844i	0.885614	+	0.464422i	$0.153738\pi$
$47$			14066.5i			0.928844i	−0.885614	+	0.464422i	$0.846262\pi$
$48$		0			0
$49$	−13151.0			−0.782473
$50$		0			0
$51$	−19092.8			−1.02789
$52$		0			0
$53$	−29222.8			−1.42900			−0.714500	−	0.699636i	$-0.753346\pi$
$53$	−29222.8			−1.42900			−0.714500	+	0.699636i	$0.753346\pi$
$54$		0			0
$55$			2435.28i			0.108553i
$56$		0			0
$57$	12566.8			0.512317
$58$		0			0
$59$	24544.2			0.917950			0.458975	−	0.888449i	$-0.348217\pi$
$59$	24544.2			0.917950			0.458975	+	0.888449i	$0.348217\pi$
$60$		0			0
$61$			29241.9i			1.00619i	0.864231	+	0.503096i	$0.167806\pi$
$61$			29241.9i			1.00619i	−0.864231	+	0.503096i	$0.832194\pi$
$62$		0			0
$63$	37889.1			1.20272
$64$		0			0
$65$	−13678.5			−0.401565
$66$		0			0
$67$	−11066.9			−0.301190			−0.150595	−	0.988596i	$-0.548119\pi$
$67$	−11066.9			−0.301190			−0.150595	+	0.988596i	$0.548119\pi$
$68$		0			0
$69$			1566.81i			0.0396182i
$70$		0			0
$71$	59435.2			1.39926			0.699629	−	0.714506i	$-0.253349\pi$
$71$	59435.2			1.39926			0.699629	+	0.714506i	$0.253349\pi$
$72$		0			0
$73$		−	38646.0i		−	0.848784i	−0.905479	−	0.424392i	$-0.860488\pi$
$73$		−	38646.0i		−	0.848784i	0.905479	−	0.424392i	$-0.139512\pi$
$74$		0			0
$75$		−	69027.9i		−	1.41700i
$76$		0			0
$77$		−	5258.06i		−	0.101064i
$78$		0			0
$79$			45902.2i			0.827495i	0.910392	+	0.413748i	$0.135780\pi$
$79$			45902.2i			0.827495i	−0.910392	+	0.413748i	$0.864220\pi$
$80$		0			0
$81$	181348.			3.07114
$82$		0			0
$83$	−118953.			−1.89531			−0.947656	−	0.319294i	$-0.896554\pi$
$83$	−118953.			−1.89531			−0.947656	+	0.319294i	$0.896554\pi$
$84$		0			0
$85$			18131.2i			0.272195i
$86$		0			0
$87$	101209.	+	87142.9i	1.43358	+	1.23434i
$88$		0			0
$89$			65646.3i			0.878486i	0.898368	+	0.439243i	$0.144753\pi$
$89$			65646.3i			0.878486i	−0.898368	+	0.439243i	$0.855247\pi$
$90$		0			0
$91$	29533.6			0.373863
$92$		0			0
$93$	99370.9			1.19138
$94$		0			0
$95$		−	11933.9i		−	0.135667i
$96$		0			0
$97$			132472.i			1.42954i	0.699362	+	0.714768i	$0.253468\pi$
$97$			132472.i			1.42954i	−0.699362	+	0.714768i	$0.746532\pi$
$98$		0			0
$99$		−	54492.4i		−	0.558790i
$100$		0			0
$101$			162419.i			1.58429i	0.610333	+	0.792145i	$0.291036\pi$
$101$			162419.i			1.58429i	−0.610333	+	0.792145i	$0.708964\pi$
$102$		0			0
$103$	−173698.			−1.61325			−0.806623	−	0.591066i	$-0.798707\pi$
$103$	−173698.			−1.61325			−0.806623	+	0.591066i	$0.798707\pi$
$104$		0			0
$105$		−	49933.8i		−	0.441999i
$106$		0			0
$107$	−85712.3			−0.723742			−0.361871	−	0.932228i	$-0.617862\pi$
$107$	−85712.3			−0.723742			−0.361871	+	0.932228i	$0.617862\pi$
$108$		0			0
$109$	−129687.			−1.04552			−0.522758	−	0.852481i	$-0.675097\pi$
$109$	−129687.			−1.04552			−0.522758	+	0.852481i	$0.675097\pi$
$110$		0			0
$111$	−247970.			−1.91026
$112$		0			0
$113$		−	52759.5i		−	0.388691i	−0.980933	−	0.194346i	$-0.937742\pi$
$113$		−	52759.5i		−	0.388691i	0.980933	−	0.194346i	$-0.0622584\pi$
$114$		0			0
$115$	1487.90			0.0104913
$116$		0			0
$117$	306075.			2.06711
$118$		0			0
$119$		−	39147.5i		−	0.253418i
$120$		0			0
$121$	153489.			0.953045
$122$		0			0
$123$	−277851.			−1.65596
$124$		0			0
$125$	−153065.			−0.876194
$126$		0			0
$127$			136355.i			0.750171i	0.926990	+	0.375086i	$0.122387\pi$
$127$			136355.i			0.750171i	−0.926990	+	0.375086i	$0.877613\pi$
$128$		0			0
$129$	433857.			2.29548
$130$		0			0
$131$			206092.i			1.04926i	0.851331	+	0.524629i	$0.175796\pi$
$131$			206092.i			1.04926i	−0.851331	+	0.524629i	$0.824204\pi$
$132$		0			0
$133$			25766.8i			0.126308i
$134$		0			0
$135$		−	316817.i		−	1.49615i
$136$		0			0
$137$		−	402381.i		−	1.83162i	−0.401610	−	0.915811i	$-0.631549\pi$
$137$		−	402381.i		−	1.83162i	0.401610	−	0.915811i	$-0.368451\pi$
$138$		0			0
$139$	378360.			1.66099			0.830497	−	0.557023i	$-0.188056\pi$
$139$	378360.			1.66099			0.830497	+	0.557023i	$0.188056\pi$
$140$		0			0
$141$	−414816.			−1.75714
$142$		0			0
$143$		−	42475.4i		−	0.173699i
$144$		0			0
$145$	82754.0	−	96111.8i	0.326865	−	0.379627i
$146$		0			0
$147$		−	387817.i		−	1.48025i
$148$		0			0
$149$	−84323.1			−0.311158			−0.155579	−	0.987823i	$-0.549724\pi$
$149$	−84323.1			−0.311158			−0.155579	+	0.987823i	$0.549724\pi$
$150$		0			0
$151$	−253711.			−0.905519			−0.452759	−	0.891633i	$-0.649560\pi$
$151$	−253711.			−0.905519			−0.452759	+	0.891633i	$0.649560\pi$
$152$		0			0
$153$		−	405710.i		−	1.40116i
$154$		0			0
$155$		−	94366.2i		−	0.315491i
$156$		0			0
$157$			377013.i			1.22070i	0.792134	+	0.610348i	$0.208970\pi$
$157$			377013.i			1.22070i	−0.792134	+	0.610348i	$0.791030\pi$
$158$		0			0
$159$		−	861766.i		−	2.70332i
$160$		0			0
$161$	−3212.56			−0.00976757
$162$		0			0
$163$			395654.i			1.16640i	0.812329	+	0.583199i	$0.198199\pi$
$163$			395654.i			1.16640i	−0.812329	+	0.583199i	$0.801801\pi$
$164$		0			0
$165$	−71815.2			−0.205355
$166$		0			0
$167$	−329959.			−0.915521			−0.457760	−	0.889076i	$-0.651348\pi$
$167$	−329959.			−0.915521			−0.457760	+	0.889076i	$0.651348\pi$
$168$		0			0
$169$	−132716.			−0.357442
$170$		0			0
$171$			267037.i			0.698362i
$172$		0			0
$173$	35163.7			0.0893262			0.0446631	−	0.999002i	$-0.485779\pi$
$173$	35163.7			0.0893262			0.0446631	+	0.999002i	$0.485779\pi$
$174$		0			0
$175$	141533.			0.349352
$176$		0			0
$177$			723798.i			1.73654i
$178$		0			0
$179$	43141.6			0.100638			0.0503192	−	0.998733i	$-0.483976\pi$
$179$	43141.6			0.100638			0.0503192	+	0.998733i	$0.483976\pi$
$180$		0			0
$181$	−17760.4			−0.0402955			−0.0201477	−	0.999797i	$-0.506414\pi$
$181$	−17760.4			−0.0402955			−0.0201477	+	0.999797i	$0.506414\pi$
$182$		0			0
$183$	−862329.			−1.90347
$184$		0			0
$185$			235481.i			0.505856i
$186$		0			0
$187$	−56302.3			−0.117739
$188$		0			0
$189$			684046.i			1.39293i
$190$		0			0
$191$		−	335097.i		−	0.664642i	−0.943166	−	0.332321i	$-0.892168\pi$
$191$		−	335097.i		−	0.664642i	0.943166	−	0.332321i	$-0.107832\pi$
$192$		0			0
$193$			21530.6i			0.0416067i	0.999784	+	0.0208034i	$0.00662240\pi$
$193$			21530.6i			0.0416067i	−0.999784	+	0.0208034i	$0.993378\pi$
$194$		0			0
$195$		−	403374.i		−	0.759663i
$196$		0			0
$197$	−236678.			−0.434503			−0.217252	−	0.976116i	$-0.569709\pi$
$197$	−236678.			−0.434503			−0.217252	+	0.976116i	$0.569709\pi$
$198$		0			0
$199$	547204.			0.979528			0.489764	−	0.871855i	$-0.337083\pi$
$199$	547204.			0.979528			0.489764	+	0.871855i	$0.337083\pi$
$200$		0			0
$201$		−	326359.i		−	0.569778i
$202$		0			0
$203$	−178676.	+	207517.i	−0.304317	+	0.353438i
$204$		0			0
$205$			263857.i			0.438515i
$206$		0			0
$207$	−33293.7			−0.0540053
$208$		0			0
$209$	37058.0			0.0586835
$210$		0			0
$211$		−	808975.i		−	1.25092i	−0.780257	−	0.625459i	$-0.784912\pi$
$211$		−	808975.i		−	1.25092i	0.780257	−	0.625459i	$-0.215088\pi$
$212$		0			0
$213$			1.75272e6i			2.64705i
$214$		0			0
$215$		−	412006.i		−	0.607866i
$216$		0			0
$217$			203748.i			0.293727i
$218$		0			0
$219$	1.13965e6			1.60569
$220$		0			0
$221$		−	316240.i		−	0.435549i
$222$		0			0
$223$	1.05341e6			1.41852			0.709258	−	0.704949i	$-0.249030\pi$
$223$	1.05341e6			1.41852			0.709258	+	0.704949i	$0.249030\pi$
$224$		0			0
$225$	1.46680e6			1.93158
$226$		0			0
$227$	777985.			1.00209			0.501045	−	0.865421i	$-0.332949\pi$
$227$	777985.			1.00209			0.501045	+	0.865421i	$0.332949\pi$
$228$		0			0
$229$		−	27916.9i		−	0.0351785i	−0.999845	−	0.0175893i	$-0.994401\pi$
$229$		−	27916.9i		−	0.0351785i	0.999845	−	0.0175893i	$-0.00559913\pi$
$230$		0			0
$231$	155058.			0.191189
$232$		0			0
$233$	18413.7			0.0222204			0.0111102	−	0.999938i	$-0.496463\pi$
$233$	18413.7			0.0222204			0.0111102	+	0.999938i	$0.496463\pi$
$234$		0			0
$235$			393924.i			0.465310i
$236$		0			0
$237$	−1.35363e6			−1.56542
$238$		0			0
$239$	850008.			0.962561			0.481281	−	0.876567i	$-0.340172\pi$
$239$	850008.			0.962561			0.481281	+	0.876567i	$0.340172\pi$
$240$		0			0
$241$	1.18696e6			1.31642			0.658210	−	0.752835i	$-0.271314\pi$
$241$	1.18696e6			1.31642			0.658210	+	0.752835i	$0.271314\pi$
$242$		0			0
$243$			2.59876e6i			2.82326i
$244$		0			0
$245$	−368285.			−0.391985
$246$		0			0
$247$			208148.i			0.217085i
$248$		0			0
$249$		−	3.50787e6i		−	3.58546i
$250$		0			0
$251$		−	1.12571e6i		−	1.12783i	−0.825834	−	0.563914i	$-0.809295\pi$
$251$		−	1.12571e6i		−	1.12783i	0.825834	−	0.563914i	$-0.190705\pi$
$252$		0			0
$253$			4620.33i			0.00453807i
$254$		0			0
$255$	−534682.			−0.514926
$256$		0			0
$257$	701816.			0.662812			0.331406	−	0.943488i	$-0.392477\pi$
$257$	701816.			0.662812			0.331406	+	0.943488i	$0.392477\pi$
$258$		0			0
$259$		−	508433.i		−	0.470960i
$260$		0			0
$261$	−1.85173e6	+	2.15062e6i	−1.68258	+	1.95417i
$262$		0			0
$263$			514282.i			0.458471i	0.973371	+	0.229236i	$0.0736226\pi$
$263$			514282.i			0.458471i	−0.973371	+	0.229236i	$0.926377\pi$
$264$		0			0
$265$	−818364.			−0.715866
$266$		0			0
$267$	−1.93588e6			−1.66188
$268$		0			0
$269$		−	1.58854e6i		−	1.33850i	−0.743039	−	0.669248i	$-0.766616\pi$
$269$		−	1.58854e6i		−	1.33850i	0.743039	−	0.669248i	$-0.233384\pi$
$270$		0			0
$271$			2.05555e6i			1.70022i	0.526609	+	0.850108i	$0.323463\pi$
$271$			2.05555e6i			1.70022i	−0.526609	+	0.850108i	$0.676537\pi$
$272$		0			0
$273$			870932.i			0.707258i
$274$		0			0
$275$		−	203554.i		−	0.162311i
$276$		0			0
$277$	1.54893e6			1.21292			0.606462	−	0.795113i	$-0.292588\pi$
$277$	1.54893e6			1.21292			0.606462	+	0.795113i	$0.292588\pi$
$278$		0			0
$279$			2.11156e6i			1.62403i
$280$		0			0
$281$	−1.04345e6			−0.788326			−0.394163	−	0.919041i	$-0.628965\pi$
$281$	−1.04345e6			−0.788326			−0.394163	+	0.919041i	$0.628965\pi$
$282$		0			0
$283$	2.23653e6			1.66000			0.829999	−	0.557764i	$-0.188341\pi$
$283$	2.23653e6			1.66000			0.829999	+	0.557764i	$0.188341\pi$
$284$		0			0
$285$	351926.			0.256649
$286$		0			0
$287$		−	569700.i		−	0.408264i
$288$		0			0
$289$	1.00067e6			0.704770
$290$		0			0
$291$	−3.90654e6			−2.70433
$292$		0			0
$293$			1.63450e6i			1.11228i	0.831088	+	0.556141i	$0.187719\pi$
$293$			1.63450e6i			1.11228i	−0.831088	+	0.556141i	$0.812281\pi$
$294$		0			0
$295$	687344.			0.459853
$296$		0			0
$297$	983800.			0.647166
$298$		0			0
$299$	−25951.6			−0.0167875
$300$		0			0
$301$			889572.i			0.565933i
$302$		0			0
$303$	−4.78967e6			−2.99709
$304$		0			0
$305$			818898.i			0.504058i
$306$		0			0
$307$		−	655372.i		−	0.396864i	−0.980115	−	0.198432i	$-0.936415\pi$
$307$		−	655372.i		−	0.396864i	0.980115	−	0.198432i	$-0.0635849\pi$
$308$		0			0
$309$		−	5.12226e6i		−	3.05187i
$310$		0			0
$311$		−	2.58280e6i		−	1.51422i	−0.653285	−	0.757112i	$-0.726610\pi$
$311$		−	2.58280e6i		−	1.51422i	0.653285	−	0.757112i	$-0.273390\pi$
$312$		0			0
$313$	−2.34852e6			−1.35498			−0.677491	−	0.735531i	$-0.736933\pi$
$313$	−2.34852e6			−1.35498			−0.677491	+	0.735531i	$0.736933\pi$
$314$		0			0
$315$	1.06106e6			0.602509
$316$		0			0
$317$		−	1.51988e6i		−	0.849497i	−0.905311	−	0.424749i	$-0.860362\pi$
$317$		−	1.51988e6i		−	0.849497i	0.905311	−	0.424749i	$-0.139638\pi$
$318$		0			0
$319$	298452.	+	256973.i	0.164210	+	0.141387i
$320$		0			0
$321$		−	2.52762e6i		−	1.36914i
$322$		0			0
$323$	275906.			0.147148
$324$		0			0
$325$	1.14333e6			0.600431
$326$		0			0
$327$		−	3.82441e6i		−	1.97786i
$328$		0			0
$329$		−	850529.i		−	0.433211i
$330$		0			0
$331$			1.80576e6i			0.905922i	0.891530	+	0.452961i	$0.149632\pi$
$331$			1.80576e6i			0.905922i	−0.891530	+	0.452961i	$0.850368\pi$
$332$		0			0
$333$		−	5.26920e6i		−	2.60396i
$334$		0			0
$335$	−309922.			−0.150883
$336$		0			0
$337$			3.01191e6i			1.44467i	0.691545	+	0.722333i	$0.256930\pi$
$337$			3.01191e6i			1.44467i	−0.691545	+	0.722333i	$0.743070\pi$
$338$		0			0
$339$	1.55585e6			0.735309
$340$		0			0
$341$	293032.			0.136467
$342$		0			0
$343$	1.81140e6			0.831342
$344$		0			0
$345$			43877.5i			0.0198470i
$346$		0			0
$347$	468309.			0.208789			0.104395	−	0.994536i	$-0.466709\pi$
$347$	468309.			0.208789			0.104395	+	0.994536i	$0.466709\pi$
$348$		0			0
$349$	333230.			0.146447			0.0732235	−	0.997316i	$-0.476671\pi$
$349$	333230.			0.146447			0.0732235	+	0.997316i	$0.476671\pi$
$350$		0			0
$351$			5.52584e6i			2.39403i
$352$		0			0
$353$	−3.93254e6			−1.67972			−0.839859	−	0.542805i	$-0.817362\pi$
$353$	−3.93254e6			−1.67972			−0.839859	+	0.542805i	$0.817362\pi$
$354$		0			0
$355$	1.66444e6			0.700967
$356$		0			0
$357$	1.15444e6			0.479404
$358$		0			0
$359$			393533.i			0.161155i	0.996748	+	0.0805776i	$0.0256765\pi$
$359$			393533.i			0.161155i	−0.996748	+	0.0805776i	$0.974324\pi$
$360$		0			0
$361$	2.29450e6			0.926659
$362$		0			0
$363$			4.52631e6i			1.80293i
$364$		0			0
$365$		−	1.08225e6i		−	0.425204i
$366$		0			0
$367$		−	1.65399e6i		−	0.641016i	−0.947246	−	0.320508i	$-0.896146\pi$
$367$		−	1.65399e6i		−	0.641016i	0.947246	−	0.320508i	$-0.103854\pi$
$368$		0			0
$369$		−	5.90415e6i		−	2.25731i
$370$		0			0
$371$	1.76695e6			0.666483
$372$		0			0
$373$	−1.96449e6			−0.731101			−0.365551	−	0.930791i	$-0.619119\pi$
$373$	−1.96449e6			−0.731101			−0.365551	+	0.930791i	$0.619119\pi$
$374$		0			0
$375$		−	4.51381e6i		−	1.65754i
$376$		0			0
$377$	−1.44337e6	+	1.67636e6i	−0.523028	+	0.607453i
$378$		0			0
$379$			742890.i			0.265660i	0.991139	+	0.132830i	$0.0424065\pi$
$379$			742890.i			0.265660i	−0.991139	+	0.132830i	$0.957594\pi$
$380$		0			0
$381$	−4.02103e6			−1.41914
$382$		0			0
$383$	−979559.			−0.341219			−0.170610	−	0.985339i	$-0.554574\pi$
$383$	−979559.			−0.341219			−0.170610	+	0.985339i	$0.554574\pi$
$384$		0			0
$385$		−	147248.i		−	0.0506289i
$386$		0			0
$387$			9.21918e6i			3.12907i
$388$		0			0
$389$			722500.i			0.242083i	0.992647	+	0.121041i	$0.0386234\pi$
$389$			722500.i			0.242083i	−0.992647	+	0.121041i	$0.961377\pi$
$390$		0			0
$391$			34399.5i			0.0113792i
$392$		0			0
$393$	−6.07755e6			−1.98494
$394$		0			0
$395$			1.28546e6i			0.414539i
$396$		0			0
$397$	−1.25012e6			−0.398086			−0.199043	−	0.979991i	$-0.563783\pi$
$397$	−1.25012e6			−0.398086			−0.199043	+	0.979991i	$0.563783\pi$
$398$		0			0
$399$	−759850.			−0.238944
$400$		0			0
$401$	−2.40314e6			−0.746307			−0.373154	−	0.927770i	$-0.621724\pi$
$401$	−2.40314e6			−0.746307			−0.373154	+	0.927770i	$0.621724\pi$
$402$		0			0
$403$			1.64591e6i			0.504828i
$404$		0			0
$405$	5.07852e6			1.53851
$406$		0			0
$407$	−731232.			−0.218811
$408$		0			0
$409$			851420.i			0.251672i	0.992051	+	0.125836i	$0.0401614\pi$
$409$			851420.i			0.251672i	−0.992051	+	0.125836i	$0.959839\pi$
$410$		0			0
$411$	1.18660e7			3.46498
$412$		0			0
$413$	−1.48406e6			−0.428130
$414$		0			0
$415$	−3.33120e6			−0.949468
$416$		0			0
$417$			1.11577e7i			3.14219i
$418$		0			0
$419$	536046.			0.149165			0.0745824	−	0.997215i	$-0.476238\pi$
$419$	536046.			0.149165			0.0745824	+	0.997215i	$0.476238\pi$
$420$		0			0
$421$			5.17342e6i			1.42257i	0.702906	+	0.711283i	$0.251886\pi$
$421$			5.17342e6i			1.42257i	−0.702906	+	0.711283i	$0.748114\pi$
$422$		0			0
$423$		−	8.81455e6i		−	2.39524i
$424$		0			0
$425$		−	1.51551e6i		−	0.406993i
$426$		0			0
$427$		−	1.76810e6i		−	0.469286i
$428$		0			0
$429$	1.25258e6			0.328596
$430$		0			0
$431$	−1.61991e6			−0.420047			−0.210024	−	0.977696i	$-0.567354\pi$
$431$	−1.61991e6			−0.420047			−0.210024	+	0.977696i	$0.567354\pi$
$432$		0			0
$433$			6.06477e6i			1.55451i	0.629183	+	0.777257i	$0.283389\pi$
$433$			6.06477e6i			1.55451i	−0.629183	+	0.777257i	$0.716611\pi$
$434$		0			0
$435$	2.83429e6	+	2.44038e6i	0.718160	+	0.618349i
$436$		0			0
$437$		−	22641.6i		−	0.00567158i
$438$		0			0
$439$	6.57652e6			1.62868			0.814339	−	0.580390i	$-0.197100\pi$
$439$	6.57652e6			1.62868			0.814339	+	0.580390i	$0.197100\pi$
$440$		0			0
$441$	8.24086e6			2.01779
$442$		0			0
$443$		−	6.21079e6i		−	1.50362i	−0.659381	−	0.751809i	$-0.729182\pi$
$443$		−	6.21079e6i		−	1.50362i	0.659381	−	0.751809i	$-0.270818\pi$
$444$		0			0
$445$			1.83838e6i			0.440083i
$446$		0			0
$447$		−	2.48665e6i		−	0.588634i
$448$		0			0
$449$			1.09530e6i			0.256399i	0.991748	+	0.128199i	$0.0409198\pi$
$449$			1.09530e6i			0.256399i	−0.991748	+	0.128199i	$0.959080\pi$
$450$		0			0
$451$	−819347.			−0.189682
$452$		0			0
$453$		−	7.48183e6i		−	1.71302i
$454$		0			0
$455$	827068.			0.187289
$456$		0			0
$457$	−4.00101e6			−0.896146			−0.448073	−	0.893997i	$-0.647889\pi$
$457$	−4.00101e6			−0.896146			−0.448073	+	0.893997i	$0.647889\pi$
$458$		0			0
$459$	7.32464e6			1.62276
$460$		0			0
$461$			1.86312e6i			0.408309i	0.978939	+	0.204154i	$0.0654445\pi$
$461$			1.86312e6i			0.408309i	−0.978939	+	0.204154i	$0.934556\pi$
$462$		0			0
$463$	−5.04837e6			−1.09446			−0.547229	−	0.836983i	$-0.684317\pi$
$463$	−5.04837e6			−1.09446			−0.547229	+	0.836983i	$0.684317\pi$
$464$		0			0
$465$	2.78281e6			0.596832
$466$		0			0
$467$			1.24533e6i			0.264236i	0.991234	+	0.132118i	$0.0421778\pi$
$467$			1.24533e6i			0.264236i	−0.991234	+	0.132118i	$0.957822\pi$
$468$		0			0
$469$	669160.			0.140475
$470$		0			0
$471$	−1.11179e7			−2.30926
$472$		0			0
$473$	1.27939e6			0.262936
$474$		0			0
$475$			997504.i			0.202853i
$476$		0			0
$477$	1.83120e7			3.68501
$478$		0			0
$479$		−	4.04718e6i		−	0.805960i	−0.915209	−	0.402980i	$-0.867974\pi$
$479$		−	4.04718e6i		−	0.805960i	0.915209	−	0.402980i	$-0.132026\pi$
$480$		0			0
$481$		−	4.10720e6i		−	0.809438i
$482$		0			0
$483$		−	94736.9i		−	0.0184778i
$484$		0			0
$485$			3.70979e6i			0.716135i
$486$		0			0
$487$	9.24469e6			1.76632			0.883161	−	0.469070i	$-0.155411\pi$
$487$	9.24469e6			1.76632			0.883161	+	0.469070i	$0.155411\pi$
$488$		0			0
$489$	−1.16677e7			−2.20654
$490$		0			0
$491$			4.51594e6i			0.845366i	0.906278	+	0.422683i	$0.138912\pi$
$491$			4.51594e6i			0.845366i	−0.906278	+	0.422683i	$0.861088\pi$
$492$		0			0
$493$	2.22205e6	+	1.91323e6i	0.411753	+	0.354527i
$494$		0			0
$495$		−	1.52602e6i		−	0.279929i
$496$		0			0
$497$	−3.59373e6			−0.652611
$498$		0			0
$499$	−16144.9			−0.00290257			−0.00145129	−	0.999999i	$-0.500462\pi$
$499$	−16144.9			−0.00290257			−0.00145129	+	0.999999i	$0.500462\pi$
$500$		0			0
$501$		−	9.73033e6i		−	1.73194i
$502$		0			0
$503$			4.29381e6i			0.756698i	0.925663	+	0.378349i	$0.123508\pi$
$503$			4.29381e6i			0.756698i	−0.925663	+	0.378349i	$0.876492\pi$
$504$		0			0
$505$			4.54844e6i			0.793660i
$506$		0			0
$507$		−	3.91372e6i		−	0.676192i
$508$		0			0
$509$	−1.02423e7			−1.75229			−0.876143	−	0.482052i	$-0.839892\pi$
$509$	−1.02423e7			−1.75229			−0.876143	+	0.482052i	$0.839892\pi$
$510$		0			0
$511$			2.33672e6i			0.395871i
$512$		0			0
$513$	−4.82105e6			−0.808813
$514$		0			0
$515$	−4.86428e6			−0.808166
$516$		0			0
$517$	−1.22324e6			−0.201272
$518$		0			0
$519$			1.03696e6i			0.168983i
$520$		0			0
$521$	2.18111e6			0.352034			0.176017	−	0.984387i	$-0.443679\pi$
$521$	2.18111e6			0.352034			0.176017	+	0.984387i	$0.443679\pi$
$522$		0			0
$523$	−2.79531e6			−0.446865			−0.223432	−	0.974719i	$-0.571726\pi$
$523$	−2.79531e6			−0.446865			−0.223432	+	0.974719i	$0.571726\pi$
$524$		0			0
$525$			4.17375e6i			0.660888i
$526$		0			0
$527$	2.18170e6			0.342190
$528$		0			0
$529$	−6.43352e6			−0.999561
$530$		0			0
$531$	−1.53802e7			−2.36715
$532$		0			0
$533$		−	4.60213e6i		−	0.701683i
$534$		0			0
$535$	−2.40032e6			−0.362563
$536$		0			0
$537$			1.27222e6i			0.190383i
$538$		0			0
$539$		−	1.14362e6i		−	0.169555i
$540$		0			0
$541$			29855.0i			0.00438554i	0.999998	+	0.00219277i	$0.000697982\pi$
$541$			29855.0i			0.00438554i	−0.999998	+	0.00219277i	$0.999302\pi$
$542$		0			0
$543$		−	523746.i		−	0.0762292i
$544$		0			0
$545$	−3.63180e6			−0.523758
$546$		0			0
$547$	−918350.			−0.131232			−0.0656160	−	0.997845i	$-0.520901\pi$
$547$	−918350.			−0.131232			−0.0656160	+	0.997845i	$0.520901\pi$
$548$		0			0
$549$		−	1.83239e7i		−	2.59470i
$550$		0			0
$551$	−1.46255e6	−	1.25928e6i	−0.205225	−	0.176703i
$552$		0			0
$553$		−	2.77546e6i		−	0.385942i
$554$		0			0
$555$	−6.94424e6			−0.956956
$556$		0			0
$557$	8.79386e6			1.20100			0.600498	−	0.799626i	$-0.294969\pi$
$557$	8.79386e6			1.20100			0.600498	+	0.799626i	$0.294969\pi$
$558$		0			0
$559$			7.18611e6i			0.972667i
$560$		0			0
$561$		−	1.66033e6i		−	0.222734i
$562$		0			0
$563$			681342.i			0.0905928i	0.998974	+	0.0452964i	$0.0144232\pi$
$563$			681342.i			0.0905928i	−0.998974	+	0.0452964i	$0.985577\pi$
$564$		0			0
$565$		−	1.47749e6i		−	0.194717i
$566$		0			0
$567$	−1.09651e7			−1.43237
$568$		0			0
$569$			1.53578e7i			1.98861i	0.106593	+	0.994303i	$0.466006\pi$
$569$			1.53578e7i			1.98861i	−0.106593	+	0.994303i	$0.533994\pi$
$570$		0			0
$571$	1.26623e6			0.162526			0.0812628	−	0.996693i	$-0.474105\pi$
$571$	1.26623e6			0.162526			0.0812628	+	0.996693i	$0.474105\pi$
$572$		0			0
$573$	9.88186e6			1.25734
$574$		0			0
$575$	−124367.			−0.0156869
$576$		0			0
$577$		−	1.08672e7i		−	1.35887i	−0.733736	−	0.679434i	$-0.762225\pi$
$577$		−	1.08672e7i		−	1.35887i	0.733736	−	0.679434i	$-0.237775\pi$
$578$		0			0
$579$	−634929.			−0.0787097
$580$		0			0
$581$	7.19246e6			0.883970
$582$		0			0
$583$		−	2.54124e6i		−	0.309652i
$584$		0			0
$585$	8.57141e6			1.03553
$586$		0			0
$587$	11692.7			0.00140061			0.000700306	−	1.00000i	$-0.499777\pi$
$587$	11692.7			0.00140061			0.000700306		1.00000i	$0.499777\pi$
$588$		0			0
$589$	−1.43598e6			−0.170554
$590$		0			0
$591$		−	6.97953e6i		−	0.821973i
$592$		0			0
$593$	1.39221e7			1.62580			0.812902	−	0.582401i	$-0.197886\pi$
$593$	1.39221e7			1.62580			0.812902	+	0.582401i	$0.197886\pi$
$594$		0			0
$595$		−	1.09630e6i		−	0.126951i
$596$		0			0
$597$			1.61368e7i			1.85303i
$598$		0			0
$599$		−	1.30384e7i		−	1.48477i	−0.669976	−	0.742383i	$-0.733696\pi$
$599$		−	1.30384e7i		−	1.48477i	0.669976	−	0.742383i	$-0.266304\pi$
$600$		0			0
$601$		−	1.20323e7i		−	1.35882i	−0.733760	−	0.679408i	$-0.762236\pi$
$601$		−	1.20323e7i		−	1.35882i	0.733760	−	0.679408i	$-0.237764\pi$
$602$		0			0
$603$	6.93491e6			0.776690
$604$		0			0
$605$	4.29835e6			0.477434
$606$		0			0
$607$			2.03807e6i			0.224516i	0.993679	+	0.112258i	$0.0358082\pi$
$607$			2.03807e6i			0.224516i	−0.993679	+	0.112258i	$0.964192\pi$
$608$		0			0
$609$	−6.11958e6	−	5.26907e6i	−0.668618	−	0.575692i
$610$		0			0
$611$		−	6.87071e6i		−	0.744559i
$612$		0			0
$613$	−1.02230e7			−1.09882			−0.549410	−	0.835553i	$-0.685147\pi$
$613$	−1.02230e7			−1.09882			−0.549410	+	0.835553i	$0.685147\pi$
$614$		0			0
$615$	−7.78103e6			−0.829563
$616$		0			0
$617$			1.04893e7i			1.10926i	0.832096	+	0.554631i	$0.187141\pi$
$617$			1.04893e7i			1.10926i	−0.832096	+	0.554631i	$0.812859\pi$
$618$		0			0
$619$		−	1.55645e7i		−	1.63271i	−0.577554	−	0.816353i	$-0.695993\pi$
$619$		−	1.55645e7i		−	1.63271i	0.577554	−	0.816353i	$-0.304007\pi$
$620$		0			0
$621$		−	601080.i		−	0.0625466i
$622$		0			0
$623$		−	3.96928e6i		−	0.409724i
$624$		0			0
$625$	3.02840e6			0.310108
$626$		0			0
$627$			1.09282e6i			0.111015i
$628$		0			0
$629$	−5.44420e6			−0.548666
$630$		0			0
$631$	8.48329e6			0.848185			0.424093	−	0.905619i	$-0.360593\pi$
$631$	8.48329e6			0.848185			0.424093	+	0.905619i	$0.360593\pi$
$632$		0			0
$633$	2.38563e7			2.36643
$634$		0			0
$635$			3.81852e6i			0.375803i
$636$		0			0
$637$	6.42353e6			0.627228
$638$		0			0
$639$	−3.72440e7			−3.60832
$640$		0			0
$641$		−	1.52658e7i		−	1.46749i	−0.679426	−	0.733744i	$-0.737771\pi$
$641$		−	1.52658e7i		−	1.46749i	0.679426	−	0.733744i	$-0.262229\pi$
$642$		0			0
$643$	7.27745e6			0.694148			0.347074	−	0.937838i	$-0.387175\pi$
$643$	7.27745e6			0.694148			0.347074	+	0.937838i	$0.387175\pi$
$644$		0			0
$645$	1.21499e7			1.14993
$646$		0			0
$647$	1.65740e6			0.155656			0.0778281	−	0.996967i	$-0.475201\pi$
$647$	1.65740e6			0.155656			0.0778281	+	0.996967i	$0.475201\pi$
$648$		0			0
$649$			2.13438e6i			0.198912i
$650$		0			0
$651$	−6.00843e6			−0.555659
$652$		0			0
$653$			6.61593e6i			0.607167i	0.952805	+	0.303583i	$0.0981831\pi$
$653$			6.61593e6i			0.607167i	−0.952805	+	0.303583i	$0.901817\pi$
$654$		0			0
$655$			5.77146e6i			0.525633i
$656$		0			0
$657$			2.42168e7i			2.18879i
$658$		0			0
$659$			1.80423e7i			1.61837i	0.587552	+	0.809186i	$0.300092\pi$
$659$			1.80423e7i			1.61837i	−0.587552	+	0.809186i	$0.699908\pi$
$660$		0			0
$661$	−5.57227e6			−0.496054			−0.248027	−	0.968753i	$-0.579782\pi$
$661$	−5.57227e6			−0.496054			−0.248027	+	0.968753i	$0.579782\pi$
$662$		0			0
$663$	9.32578e6			0.823951
$664$		0			0
$665$			721581.i			0.0632748i
$666$		0			0
$667$	157005.	−	182348.i	0.0136647	−	0.0158703i
$668$		0			0
$669$			3.10645e7i			2.68349i
$670$		0			0
$671$	−2.54289e6			−0.218033
$672$		0			0
$673$	−1.03563e7			−0.881391			−0.440696	−	0.897657i	$-0.645268\pi$
$673$	−1.03563e7			−0.881391			−0.440696	+	0.897657i	$0.645268\pi$
$674$		0			0
$675$			2.64813e7i			2.23707i
$676$		0			0
$677$		−	4.86050e6i		−	0.407576i	−0.979015	−	0.203788i	$-0.934675\pi$
$677$		−	4.86050e6i		−	0.407576i	0.979015	−	0.203788i	$-0.0653254\pi$
$678$		0			0
$679$		−	8.00988e6i		−	0.666733i
$680$		0			0
$681$			2.29424e7i			1.89571i
$682$		0			0
$683$	1.76295e7			1.44607			0.723033	−	0.690813i	$-0.242747\pi$
$683$	1.76295e7			1.44607			0.723033	+	0.690813i	$0.242747\pi$
$684$		0			0
$685$		−	1.12684e7i		−	0.917563i
$686$		0			0
$687$	823255.			0.0665492
$688$		0			0
$689$	1.42737e7			1.14548
$690$		0			0
$691$	1.58720e6			0.126455			0.0632274	−	0.997999i	$-0.479861\pi$
$691$	1.58720e6			0.126455			0.0632274	+	0.997999i	$0.479861\pi$
$692$		0			0
$693$			3.29487e6i			0.260618i
$694$		0			0
$695$	1.05957e7			0.832086
$696$		0			0
$697$	−6.10024e6			−0.475625
$698$		0			0
$699$			543012.i			0.0420355i
$700$		0			0
$701$	−7.72360e6			−0.593642			−0.296821	−	0.954933i	$-0.595926\pi$
$701$	−7.72360e6			−0.593642			−0.296821	+	0.954933i	$0.595926\pi$
$702$		0			0
$703$	3.58336e6			0.273465
$704$		0			0
$705$	−1.16166e7			−0.880253
$706$		0			0
$707$		−	9.82064e6i		−	0.738910i
$708$		0			0
$709$	−1.63586e7			−1.22216			−0.611082	−	0.791567i	$-0.709266\pi$
$709$	−1.63586e7			−1.22216			−0.611082	+	0.791567i	$0.709266\pi$
$710$		0			0
$711$		−	2.87638e7i		−	2.13389i
$712$		0			0
$713$		−	179036.i		−	0.0131892i
$714$		0			0
$715$		−	1.18950e6i		−	0.0870157i
$716$		0			0
$717$			2.50663e7i			1.82093i
$718$		0			0
$719$	−1.08103e7			−0.779857			−0.389928	−	0.920845i	$-0.627500\pi$
$719$	−1.08103e7			−0.779857			−0.389928	+	0.920845i	$0.627500\pi$
$720$		0			0
$721$	1.05026e7			0.752415
$722$		0			0
$723$			3.50030e7i			2.49034i
$724$		0			0
$725$	−6.91704e6	+	8.03357e6i	−0.488737	+	0.567627i
$726$		0			0
$727$		−	7.87081e6i		−	0.552310i	−0.961113	−	0.276155i	$-0.910940\pi$
$727$		−	7.87081e6i		−	0.552310i	0.961113	−	0.276155i	$-0.0890604\pi$
$728$		0			0
$729$	−3.25688e7			−2.26978
$730$		0			0
$731$	9.52537e6			0.659308
$732$		0			0
$733$		−	1.93005e7i		−	1.32681i	−0.748260	−	0.663405i	$-0.769111\pi$
$733$		−	1.93005e7i		−	1.32681i	0.748260	−	0.663405i	$-0.230889\pi$
$734$		0			0
$735$		−	1.08606e7i		−	0.741539i
$736$		0			0
$737$		−	962390.i		−	0.0652653i
$738$		0			0
$739$			4.80065e6i			0.323362i	0.986843	+	0.161681i	$0.0516916\pi$
$739$			4.80065e6i			0.323362i	−0.986843	+	0.161681i	$0.948308\pi$
$740$		0			0
$741$	−6.13820e6			−0.410672
$742$		0			0
$743$		−	2.45653e7i		−	1.63249i	−0.577708	−	0.816243i	$-0.696053\pi$
$743$		−	2.45653e7i		−	1.63249i	0.577708	−	0.816243i	$-0.303947\pi$
$744$		0			0
$745$	−2.36141e6			−0.155877
$746$		0			0
$747$	7.45399e7			4.88751
$748$		0			0
$749$	5.18257e6			0.337552
$750$		0			0
$751$			6.46351e6i			0.418185i	0.977896	+	0.209093i	$0.0670510\pi$
$751$			6.46351e6i			0.418185i	−0.977896	+	0.209093i	$0.932949\pi$
$752$		0			0
$753$	3.31967e7			2.13357
$754$		0			0
$755$	−7.10501e6			−0.453625
$756$		0			0
$757$		−	1.37372e7i		−	0.871279i	−0.900121	−	0.435639i	$-0.856522\pi$
$757$		−	1.37372e7i		−	0.871279i	0.900121	−	0.435639i	$-0.143478\pi$
$758$		0			0
$759$	−136251.			−0.00858492
$760$		0			0
$761$	2.41213e7			1.50987			0.754934	−	0.655801i	$-0.227669\pi$
$761$	2.41213e7			1.50987			0.754934	+	0.655801i	$0.227669\pi$
$762$		0			0
$763$	7.84149e6			0.487627
$764$		0			0
$765$		−	1.13616e7i		−	0.701919i
$766$		0			0
$767$	−1.19885e7			−0.735827
$768$		0			0
$769$		−	2.36696e7i		−	1.44336i	−0.692226	−	0.721680i	$-0.743370\pi$
$769$		−	2.36696e7i		−	1.44336i	0.692226	−	0.721680i	$-0.256630\pi$
$770$		0			0
$771$			2.06962e7i			1.25388i
$772$		0			0
$773$			2.34254e7i			1.41006i	0.709177	+	0.705031i	$0.249067\pi$
$773$			2.34254e7i			1.41006i	−0.709177	+	0.705031i	$0.750933\pi$
$774$		0			0
$775$			7.88766e6i			0.471730i
$776$		0			0
$777$	1.49934e7			0.890941
$778$		0			0
$779$	4.01516e6			0.237060
$780$		0			0
$781$			5.16853e6i			0.303207i
$782$		0			0
$783$	−3.88271e7	−	3.34308e7i	−2.26324	−	1.94869i
$784$		0			0
$785$			1.05580e7i			0.611515i
$786$		0			0
$787$	−1.12715e7			−0.648700			−0.324350	−	0.945937i	$-0.605146\pi$
$787$	−1.12715e7			−0.648700			−0.324350	+	0.945937i	$0.605146\pi$
$788$		0			0
$789$	−1.51659e7			−0.867315
$790$		0			0
$791$			3.19009e6i			0.181285i
$792$		0			0
$793$		−	1.42830e7i		−	0.806560i
$794$		0			0
$795$		−	2.41332e7i		−	1.35424i
$796$		0			0
$797$			1.47100e7i			0.820291i	0.912020	+	0.410145i	$0.134522\pi$
$797$			1.47100e7i			0.820291i	−0.912020	+	0.410145i	$0.865478\pi$
$798$		0			0
$799$	−9.10731e6			−0.504688
$800$		0			0
$801$		−	4.11361e7i		−	2.26538i
$802$		0			0
$803$	3.36068e6			0.183924
$804$		0			0
$805$	−89965.5			−0.00489313
$806$		0			0
$807$	4.68453e7			2.53211
$808$		0			0
$809$		−	2.15758e7i		−	1.15903i	−0.814961	−	0.579515i	$-0.803242\pi$
$809$		−	2.15758e7i		−	1.15903i	0.814961	−	0.579515i	$-0.196758\pi$
$810$		0			0
$811$	−1.00894e7			−0.538659			−0.269330	−	0.963048i	$-0.586802\pi$
$811$	−1.00894e7			−0.538659			−0.269330	+	0.963048i	$0.586802\pi$
$812$		0			0
$813$	−6.06171e7			−3.21639
$814$		0			0
$815$			1.10800e7i			0.584315i
$816$		0			0
$817$	−6.26956e6			−0.328611
$818$		0			0
$819$	−1.85067e7			−0.964095
$820$		0			0
$821$	−3.20138e6			−0.165760			−0.0828798	−	0.996560i	$-0.526412\pi$
$821$	−3.20138e6			−0.165760			−0.0828798	+	0.996560i	$0.526412\pi$
$822$		0			0
$823$		−	3.82993e7i		−	1.97102i	−0.169621	−	0.985509i	$-0.554254\pi$
$823$		−	3.82993e7i		−	1.97102i	0.169621	−	0.985509i	$-0.445746\pi$
$824$		0			0
$825$	6.00272e6			0.307053
$826$		0			0
$827$		−	3.94368e6i		−	0.200511i	−0.994962	−	0.100255i	$-0.968034\pi$
$827$		−	3.94368e6i		−	0.200511i	0.994962	−	0.100255i	$-0.0319660\pi$
$828$		0			0
$829$			2.10818e7i			1.06542i	0.846298	+	0.532710i	$0.178827\pi$
$829$			2.10818e7i			1.06542i	−0.846298	+	0.532710i	$0.821173\pi$
$830$		0			0
$831$			4.56773e7i			2.29455i
$832$		0			0
$833$		−	8.51456e6i		−	0.425157i
$834$		0			0
$835$	−9.24026e6			−0.458636
$836$		0			0
$837$	−3.81219e7			−1.88088
$838$		0			0
$839$		−	9.75710e6i		−	0.478538i	−0.970953	−	0.239269i	$-0.923092\pi$
$839$		−	9.75710e6i		−	0.478538i	0.970953	−	0.239269i	$-0.0769077\pi$
$840$		0			0
$841$	−3.04658e6	−	2.02836e7i	−0.148533	−	0.988907i
$842$		0			0
$843$		−	3.07708e7i		−	1.49132i
$844$		0			0
$845$	−3.71661e6			−0.179063
$846$		0			0
$847$	−9.28065e6			−0.444498
$848$		0			0
$849$			6.59541e7i			3.14031i
$850$		0			0
$851$			446767.i			0.0211474i
$852$		0			0
$853$			6.34997e6i			0.298813i	0.988776	+	0.149406i	$0.0477363\pi$
$853$			6.34997e6i			0.298813i	−0.988776	+	0.149406i	$0.952264\pi$
$854$		0			0
$855$			7.47818e6i			0.349849i
$856$		0			0
$857$	1.12338e7			0.522487			0.261243	−	0.965273i	$-0.415868\pi$
$857$	1.12338e7			0.522487			0.261243	+	0.965273i	$0.415868\pi$
$858$		0			0
$859$			3.46530e7i			1.60235i	0.598428	+	0.801177i	$0.295792\pi$
$859$			3.46530e7i			1.60235i	−0.598428	+	0.801177i	$0.704208\pi$
$860$		0			0
$861$	1.68002e7			0.772336
$862$		0			0
$863$	−8.92450e6			−0.407903			−0.203952	−	0.978981i	$-0.565378\pi$
$863$	−8.92450e6			−0.407903			−0.203952	+	0.978981i	$0.565378\pi$
$864$		0			0
$865$	984735.			0.0447486
$866$		0			0
$867$			2.95094e7i			1.33325i
$868$		0			0
$869$	−3.99169e6			−0.179311
$870$		0			0
$871$	5.40558e6			0.241433
$872$		0			0
$873$		−	8.30113e7i		−	3.68639i
$874$		0			0
$875$	9.25502e6			0.408655
$876$		0			0
$877$	−1.67905e7			−0.737166			−0.368583	−	0.929595i	$-0.620157\pi$
$877$	−1.67905e7			−0.737166			−0.368583	+	0.929595i	$0.620157\pi$
$878$		0			0
$879$	−4.82006e7			−2.10417
$880$		0			0
$881$			3.44228e6i			0.149419i	0.997205	+	0.0747095i	$0.0238030\pi$
$881$			3.44228e6i			0.149419i	−0.997205	+	0.0747095i	$0.976197\pi$
$882$		0			0
$883$	1.49330e7			0.644533			0.322266	−	0.946649i	$-0.395555\pi$
$883$	1.49330e7			0.644533			0.322266	+	0.946649i	$0.395555\pi$
$884$		0			0
$885$			2.02695e7i			0.869929i
$886$		0			0
$887$			4.10323e7i			1.75112i	0.483106	+	0.875562i	$0.339508\pi$
$887$			4.10323e7i			1.75112i	−0.483106	+	0.875562i	$0.660492\pi$
$888$		0			0
$889$		−	8.24464e6i		−	0.349878i
$890$		0			0
$891$			1.57701e7i			0.665489i
$892$		0			0
$893$	5.99440e6			0.251546
$894$		0			0
$895$	1.20815e6			0.0504154
$896$		0			0
$897$		−	765300.i		−	0.0317578i
$898$		0			0
$899$	−1.15649e7	−	9.95761e6i	−0.477248	−	0.410919i
$900$		0			0
$901$		−	1.89201e7i		−	0.776448i
$902$		0			0
$903$	−2.62331e7			−1.07061
$904$		0			0
$905$	−497368.			−0.0201863
$906$		0			0
$907$			2.01708e7i			0.814149i	0.913395	+	0.407074i	$0.133451\pi$
$907$			2.01708e7i			0.814149i	−0.913395	+	0.407074i	$0.866549\pi$
$908$		0			0
$909$		−	1.01777e8i		−	4.08546i
$910$		0			0
$911$			1.50765e7i			0.601874i	0.953644	+	0.300937i	$0.0972995\pi$
$911$			1.50765e7i			0.601874i	−0.953644	+	0.300937i	$0.902701\pi$
$912$		0			0
$913$		−	1.03443e7i		−	0.410698i
$914$		0			0
$915$	−2.41489e7			−0.953554
$916$		0			0
$917$		−	1.24613e7i		−	0.489372i
$918$		0			0
$919$	−2.03024e7			−0.792972			−0.396486	−	0.918041i	$-0.629771\pi$
$919$	−2.03024e7			−0.792972			−0.396486	+	0.918041i	$0.629771\pi$
$920$		0			0
$921$	1.93266e7			0.750769
$922$		0			0
$923$	−2.90308e7			−1.12164
$924$		0			0
$925$		−	1.96829e7i		−	0.756369i
$926$		0			0
$927$	1.08845e8			4.16014
$928$		0			0
$929$	−3.46473e6			−0.131714			−0.0658568	−	0.997829i	$-0.520978\pi$
$929$	−3.46473e6			−0.131714			−0.0658568	+	0.997829i	$0.520978\pi$
$930$		0			0
$931$			5.60425e6i			0.211906i
$932$		0			0
$933$	7.61657e7			2.86454
$934$		0			0
$935$	−1.57671e6			−0.0589823
$936$		0			0
$937$	−4.34253e7			−1.61582			−0.807912	−	0.589304i	$-0.799402\pi$
$937$	−4.34253e7			−1.61582			−0.807912	+	0.589304i	$0.799402\pi$
$938$		0			0
$939$		−	6.92567e7i		−	2.56329i
$940$		0			0
$941$	2.01185e7			0.740663			0.370331	−	0.928900i	$-0.379244\pi$
$941$	2.01185e7			0.740663			0.370331	+	0.928900i	$0.379244\pi$
$942$		0			0
$943$			500603.i			0.0183322i
$944$		0			0
$945$			1.91562e7i			0.697799i
$946$		0			0
$947$			2.27622e7i			0.824782i	0.911007	+	0.412391i	$0.135306\pi$
$947$			2.27622e7i			0.824782i	−0.911007	+	0.412391i	$0.864694\pi$
$948$		0			0
$949$			1.88764e7i			0.680383i
$950$		0			0
$951$	4.48206e7			1.60704
$952$		0			0
$953$	−3.46041e7			−1.23423			−0.617114	−	0.786874i	$-0.711698\pi$
$953$	−3.46041e7			−1.23423			−0.617114	+	0.786874i	$0.711698\pi$
$954$		0			0
$955$		−	9.38417e6i		−	0.332956i
$956$		0			0
$957$	−7.57801e6	+	8.80122e6i	−0.267470	+	0.310644i
$958$		0			0
$959$			2.43298e7i			0.854265i
$960$		0			0
$961$	1.72743e7			0.603380
$962$		0			0
$963$	5.37101e7			1.86634
$964$		0			0
$965$			602951.i			0.0208432i
$966$		0			0
$967$			3.36033e7i			1.15562i	0.816171	+	0.577811i	$0.196093\pi$
$967$			3.36033e7i			1.15562i	−0.816171	+	0.577811i	$0.803907\pi$
$968$		0			0
$969$			8.13633e6i			0.278368i
$970$		0			0
$971$			8.64742e6i			0.294333i	0.989112	+	0.147166i	$0.0470152\pi$
$971$			8.64742e6i			0.294333i	−0.989112	+	0.147166i	$0.952985\pi$
$972$		0			0
$973$	−2.28774e7			−0.774685
$974$		0			0
$975$			3.37162e7i			1.13587i
$976$		0			0
$977$	4.76115e7			1.59579			0.797894	−	0.602798i	$-0.205947\pi$
$977$	4.76115e7			1.59579			0.797894	+	0.602798i	$0.205947\pi$
$978$		0			0
$979$	−5.70865e6			−0.190360
$980$		0			0
$981$	8.12662e7			2.69611
$982$		0			0
$983$		−	5.30624e6i		−	0.175147i	−0.996158	−	0.0875735i	$-0.972089\pi$
$983$		−	5.30624e6i		−	0.175147i	0.996158	−	0.0875735i	$-0.0279113\pi$
$984$		0			0
$985$	−6.62801e6			−0.217667
$986$		0			0
$987$	2.50817e7			0.819529
$988$		0			0
$989$		−	781679.i		−	0.0254119i
$990$		0			0
$991$	−1.33306e7			−0.431187			−0.215593	−	0.976483i	$-0.569169\pi$
$991$	−1.33306e7			−0.431187			−0.215593	+	0.976483i	$0.569169\pi$
$992$		0			0
$993$	−5.32511e7			−1.71378
$994$		0			0
$995$	1.53241e7			0.490701
$996$		0			0
$997$			1.90595e7i			0.607260i	0.952790	+	0.303630i	$0.0981987\pi$
$997$			1.90595e7i			0.607260i	−0.952790	+	0.303630i	$0.901801\pi$
$998$		0			0
$999$	9.51295e7			3.01579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.6.c.a.57.12	yes	12
3.2	odd	2		1044.6.h.a.289.5		12
4.3	odd	2		464.6.e.a.289.1		12
29.28	even	2	inner	116.6.c.a.57.1	✓	12
87.86	odd	2		1044.6.h.a.289.6		12
116.115	odd	2		464.6.e.a.289.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.6.c.a.57.1	✓	12	29.28	even	2	inner
116.6.c.a.57.12	yes	12	1.1	even	1	trivial
464.6.e.a.289.1		12	4.3	odd	2
464.6.e.a.289.12		12	116.115	odd	2
1044.6.h.a.289.5		12	3.2	odd	2
1044.6.h.a.289.6		12	87.86	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	116.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+29.4895i q^{3} +28.0043 q^{5} -60.4647 q^{7} -626.633 q^{9} +O(q^{10})\)	\(q+29.4895i q^{3} +28.0043 q^{5} -60.4647 q^{7} -626.633 q^{9} +86.9608i q^{11} -488.444 q^{13} +825.834i q^{15} +647.445i q^{17} -426.146i q^{19} -1783.08i q^{21} +53.1312 q^{23} -2340.76 q^{25} -11313.1i q^{27} +(2955.04 - 3432.04i) q^{29} -3369.70i q^{31} -2564.43 q^{33} -1693.27 q^{35} +8408.75i q^{37} -14404.0i q^{39} +9422.03i q^{41} -14712.3i q^{43} -17548.4 q^{45} +14066.5i q^{47} -13151.0 q^{49} -19092.8 q^{51} -29222.8 q^{53} +2435.28i q^{55} +12566.8 q^{57} +24544.2 q^{59} +29241.9i q^{61} +37889.1 q^{63} -13678.5 q^{65} -11066.9 q^{67} +1566.81i q^{69} +59435.2 q^{71} -38646.0i q^{73} -69027.9i q^{75} -5258.06i q^{77} +45902.2i q^{79} +181348. q^{81} -118953. q^{83} +18131.2i q^{85} +(101209. + 87142.9i) q^{87} +65646.3i q^{89} +29533.6 q^{91} +99370.9 q^{93} -11933.9i q^{95} +132472. i q^{97} -54492.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9}+O(q^{10}) \) 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9	\( 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9} - 194 q^{13} + 3996 q^{23} + 1782 q^{25} - 1900 q^{29} + 7266 q^{33} + 12348 q^{35} - 13988 q^{45} - 8396 q^{49} - 10788 q^{51} - 28626 q^{53} + 11140 q^{57} + 27356 q^{59} + 55392 q^{63} - 55126 q^{65} - 13184 q^{67} + 44352 q^{71} + 74264 q^{81} - 192628 q^{83} + 98908 q^{87} + 13580 q^{91} + 59710 q^{93}+O(q^{100}) \) 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9 - 194 * q^13 + 3996 * q^23 + 1782 * q^25 - 1900 * q^29 + 7266 * q^33 + 12348 * q^35 - 13988 * q^45 - 8396 * q^49 - 10788 * q^51 - 28626 * q^53 + 11140 * q^57 + 27356 * q^59 + 55392 * q^63 - 55126 * q^65 - 13184 * q^67 + 44352 * q^71 + 74264 * q^81 - 192628 * q^83 + 98908 * q^87 + 13580 * q^91 + 59710 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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